Standard generators of L2(223) are a of
order 2, b of order 3 such that ab has order 223.
Standard generators of the double cover
2.L2(223) = SL2(223) are preimages
A and B such that B has order 3 and
AB has order 223.

Standard generators of L2(223):2 = PGL2(223) are not yet defined.
Standard generators of the double cover 2.L2(223):2 = 2.PGL2(223) are not yet defined.